A077779 -id:A077779 - cp's OEIS frontend

A107123 Numbers k such that (10^(2k+1)+1810^k-1)/9 is prime.

0, 1, 2, 19, 97, 9818

Offset: 1

Farideh Firoozbakht, May 19 2005

A number k is in the sequence iff the palindromic number 1(k).3.1(k) is prime (1(k) means k copies of 1; dot between numbers means concatenation). If k is a positive term of the sequence then k is not of the form 3m, 6m+4, 12m+10, 28m+5, 28m+8, etc. (the proof is easy).
The palindromic number 1(k).2.1(k) is never prime for k > 0 because it is (1.0(k-1).1)*(1(k+1)). - Robert Israel, Jun 11 2015
a(7) > 10^5. - Robert Price, Apr 02 2016

			19 is in the sequence because the palindromic number (10^(2*19+1)+18*10^19-1)/9 = 1(19).3.1(19) = 111111111111111111131111111111111111111 is prime.

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 11...11311...11
Index entries for primes involving repunits.

Cf. A004023, A077775-A077798, A107124, A107125, A107126, A107127, A107648, A107649, A115073, A183174-A183187.

select(n -> isprime((10^(2*n+1)+18*10^n-1)/9), [$0..100]); # Robert Israel, Jun 11 2015

Do[If[PrimeQ[(10^(2n + 1) + 18*10^n - 1)/9], Print[n]], {n, 2500}]

for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+18*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077779(n-1)-1)/2, for n > 1. [Corrected by M. F. Hasler, Feb 06 2020]

Edited by Ray Chandler, Dec 28 2010

A332113 a(n) = (10^(2n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

3, 131, 11311, 1113111, 111131111, 11111311111, 1111113111111, 111111131111111, 11111111311111111, 1111111113111111111, 111111111131111111111, 11111111111311111111111, 1111111111113111111111111, 111111111111131111111111111, 11111111111111311111111111111, 1111111111111113111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107123 = {0, 1, 2, 19, 97, 9818, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (1)3(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11311...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077779-1)/2 = A107123: indices of primes; A331864 & A331865 (non-palindromic variants).
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332123 .. A332193 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332113 := n -> (10^(2*n+1)-1)/9+2*10^n;

Array[(10^(2 # + 1)-1)/9 + 2*10^# &, 15, 0]

apply( {A332113(n)=10^(n*2+1)\9+2*10^n}, [0..15])

def A332113(n): return 10**(n*2+1)//9+2*10**n

a(n) = A138148(n) + 3*10^n = A002275(2n+1) + 2*10^n.
G.f.: (3 - 202*x + 100*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A331865 Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 12, 20, 39, 74, 78, 80, 84, 104, 195, 654, 980, 2076, 5940, 19637

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 2*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 3, and again floor(n/2) digits 1 (except for n=0). For odd n, this is a palindrome (a.k.a. wing prime, cf. A077779), for even n the digit 3 is just left to the middle of the number.
a(22) > 50000. - Michael S. Branicky, Feb 19 2025

			For n = 0, R(0) + 2*10^floor(0/2) = 2 is prime.
For n = 1, R(1) + 2*10^floor(1/2) = 3 is prime.
For n = 2, R(2) + 2*10^floor(2/2) = 31 is prime.
For n = 3, R(3) + 2*10^floor(3/2) = 131 is prime.
For n = 5, R(5) + 2*10^floor(5/2) = 11311 is prime.
For n = 6, R(6) + 2*10^floor(6/2) = 113111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331860 & A331863 (variants with digit 2 resp. 0 instead of 3), A331864 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077779 (odd terms).

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 2*10^Floor[#/2]] &] (* Michael De Vlieger, Jan 31 2020 *)

for(n=0,9999,isprime(p=10^n\9+2*10^(n\2))&&print1(n","))

a(18)-a(20) from Giovanni Resta, Jan 30 2020

a(21) from Michael S. Branicky, Feb 19 2025

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A107123 Numbers k such that (10^(2*k+1)+18*10^k-1)/9 is prime.

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A332113 a(n) = (10^(2n+1)-1)/9 + 2*10^n.

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A331865 Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

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A107123 Numbers k such that (10^(2k+1)+1810^k-1)/9 is prime.